Question

确定a,b的值,使得当x→0时,f(x)=x-(a+bcosx)sinx成为x^5的同价无穷小量

Answer 1

分别将cosx与sinx在0点泰勒展开.

因为只要考虑x^5的同阶无穷小量,根据原式,我们只需作如下近似展开：

cosx=1-x^2/2+x^4/24+o(x^4)

sinx=x-x^3/6+x^5/120+o(x^5)

代入原式,我们有

f(x)=x-[a+b(1-x^2/2+x^4/24+o(x^4))]*[x-x^3/6+x^5/120+o(x^5)]

=(1-a-b)*x - (a/6 + 2b/3)*x^3 + (a/120 + 2b/15)*x^5 + o(x^5)

所以,1-a-b=0,a/6 + 2b/3=0,a/120 + 2b/15≠0.

解得,a=4/3,b=-1/3.